4. Results and discussion
In Sect. 4.1, we present the measured scattering matrices as functions of the scattering angle for the samples studied. We compare the experimental results for the different samples in Sect. 4.2. Furthermore, the measured angular distributions of the degree of linear polarization for unpolarized incident light is compared with observational data of comets and asteroids in Sect. 4.3.
In Fig. 5 and Fig. 6, we present the complete scattering matrices for the Allende meteorite particles (stars) and the olivine samples XL (circles), L (squares), M (diamonds), and S (triangles) at 442 nm and 633 nm, respectively. We refrained from showing the element ratios , , and , since they were found to be zero over the entire range of scattering angles within the accuracy of the measurements. This is in agreement with the assumption of randomly oriented particles with equal amounts of particles and their mirror particles (Van de Hulst 1957). Clearly, Eq. (2) is valid for our samples. The or () is shown on a logarithmic scale and normalized to 1 at 30 degrees. The measurements for olivine sample XL show larger error bars than the measurements for the other samples. This is predominantly due to the fact that the particles in sample XL are relatively large so that relatively few particles are present in the scattering volume during the measurements, thereby decreasing the signal-to-noise ratio. An increase in the jet flow would have improved the accuracy, but this was not possible because of the limited amount of sample material available.
In all cases the , , are smooth functions of the scattering angle showing a strong forward peak; they are featureless and flat at side scattering angles and have almost no structure at back-scattering angles. This behavior seems to be a general property of ensembles of natural mineral particles (Jaggard et al. 1981; West et al. 1997; Volten et al. 2000). Looking in more detail, we see some differences between the four olivine samples. Although the shapes of the curves are similar their steepnesses (see Table 3), defined as the measured maximum of divided by its measured minimum, over the scattering angle range of 5o to 173o , are different. The smallest value of the steepness at both wavelengths is presented by olivine sample L. In contrast, the Allende meteorite sample (which consists of the smallest particles) exhibits the largest steepness at 442 nm, while its value at 633 nm is quite low. This indicates that the complex refractive index strongly influences the steepness, because for Allende meteorite we expect a larger imaginary part of the refractive index due to the high percentage of iron in the sample (see Sect. 3.4).
Table 3. Steepness of
The measured curves show only minor differences for the four olivine samples (XL, L, M and S) at 442 nm. However, at 633 nm more pronounced differences occur and the highest maximum values are obtained for olivine samples M and S. We will discuss the results for this function in more detail in Sect. 4.3.
The ratios are often used as a measure for the nonsphericity of the particles, since for spheres this function is equal to 1 at all scattering angles. In all the samples we have studied in this work, this ratio decreases from almost 1 at angles close to the forward direction to a minimum at side-scattering, and increases again at back-scattering angles (Fig. 5 and Fig. 6).
The general pattern of is the same for all the samples with a broad side-scattering maximum separating two negative branches at small and large scattering angles.
4.2. Comparison of different samples
In Fig. 5 and Fig. 6, we see that the measurements for olivine samples M and S (diamonds and triangles respectively) yield nearly the same functions at both wavelengths. If we compare the results of and for these two samples (M and S) with those for the other two olivine samples (XL and L) we see large differences. For these ratios of scattering matrix elements, samples S and M exhibit larger values at most scattering angles than the other two olivine samples. Indeed, when looking at the values for in Table 2, it is surprising that samples S and M show such a similar scattering behavior while samples L and M, that show a similar difference in values, differ highly in scattering behavior.
Another interesting feature is that for almost all scattering angles the lowest values for and occur for olivine sample L. Since sample L is intermediate in size (see Table 2), we do not a priori expect this sample to show this extreme behavior. It is remarkable that the XL sample, while having the largest , presents scattering behavior that is intermediate with respect to L and M. The reason might be that is larger for sample XL than for L, so that the small particles in sample XL contribute more to the total scattering than those in sample L. The projected surface distributions shown in Fig. 3 support this argument.
4.3. Measured degree of linear polarization compared with data for comets and asteroids
In Fig. 7, we compare at 442 nm and 633 nm for olivine sample S (left panel) and for the Allende meteorite (right panel). The measured at scattering angles between about 45o and 145o for the olivine sample is higher at 633 nm than at 442 nm. However, for the Allende sample, the curves are quite similar at both wavelengths and at almost all scattering angles. The behavior of the maximum of for irregular particles may be clarified along the following lines.
We first consider some rules that are based on limiting cases for very small particles and very large particles. For very small particles (sizes smaller than or approximately equal to the wavelength) the maximum polarization tends to decrease with the size parameter, and, therefore, increase with wavelength if the refractive index m is constant (Yanamandra-Fisher & Hanner 1999; Mishchenko et al. 2000). For very large particles (sizes much larger than the wavelength) we assume that geometric optics holds. Then, the behavior of the maximum polarization as a function of wavelength will depend on the product of the absorption coefficient, a, and the average diameter, d, of the particles, because this product determines the contribution of internally reflected light to the scattered light. If the product ad is small, many internal reflections occur which will lower the maximum degree of polarization. The absorption coefficient is related to the imaginary part of the refractive index, k, and the wavelength, , as follows.
For the olivine particles, which have a low iron content, k is small and many internal reflections are expected for all particles that have large radii in Fig. 3. In contrast, the Allende particles have a high iron content and the value of k is higher, particularly at 442 nm. It then depends on the ratio of d and how strongly internal reflections will contribute to the scattered light. This illustrates that, even in the limit of geometric optics, it is difficult to predict what will happen with the maximum degree of polarization as a function of wavelength and/or size, in particular since k itself is a function of wavelength.
Fig. 3 suggests that most of the scattering by olivine and Allende particles originates from small particles, because their projected surface area is relatively large. However, very small particles are inefficient scatterers, which makes it difficult to estimate precisely the relative contribution of small, intermediate and large particles to the total scattering. Here small particles refer to Rayleigh-like behavior and large particles refer to particles that show geometric-optics-like behavior.
These considerations lead to the conclusion that we cannot fully interpret the results of the measurements. For such an interpretation theoretical calculations using advanced methods that yield scattering matrices of irregular mineral particles for small, intermediate, and large particles are required. Such calculations, if at all possible, are beyond the scope of this paper. However, some clarifications seem possible, based on the rules that hold for the limiting cases and were mentioned above.
Although the clarifications given above seem satisfactory, calculations are needed to confirm them. Also, we would expect that the polarization of the dark Allende particles would be larger than those of the light olivine particles, because small particles tend to show more polarization if the absorption increases (see e.g. Wielaard et al., 1997). However, that is not evident in the measured polarization curves. As yet we cannot explain the low polarization of the Allende sample as compared to the olivine samples.
Another interesting feature (see top right panels of Fig. 5 and Fig. 6) is that for scattering angles close to the backward direction the degree of linear polarization becomes negative. Such negative polarization has also been observed in a variety of Solar System bodies such as asteroids and comets. In order to compare with the polarization data obtained for comets we use the phase angle , instead of the scattering angle , (). Furthermore, we write . In Table 4, we present the measured main parameters of the curve of degree of polarization vs. phase angle in the region of minimum polarization ( , ) , inversion (, h) where h is the slope of the positive branch of P at , and maximum polarization ( , ). These parameters are marked on a simulated polarization vs. phase curve in Fig. 8.
Table 4. Measured polarization parameters.
The polarization data of asteroids (Dollfus 1989) and comets (Levasseur-Regourd et al. 1996) obtained by different groups of observers are shown in Table 5. Generally, and are larger in Table 4 compared to Table 5. Further, h and in Table 4 are similar to h and in Table 5. The observational data show little difference for phase angles below . For larger angles, comets with a maximum in polarization of and of are found (see Table 5). As discussed earlier, according to our measured results, the maximum of is directly related with the size of the particles. Within the range of sizes of our samples, the smaller the size parameter, the higher the maximum of . Therefore, the differences in found for different comets could indicate differences in the size distributions of the cometary particles. Furthermore, according to the measured for Allende meteorite and the discussion given above, the color of the particles of different comets can also produce differences in their .
It has been suggested that the value of the inversion angle could be a diagnostic of the texture of the particles. Observational data of P/Halley indicate that is smaller for positions close to the nucleus () than for the inner coma () (Dollfus 1989). Dollfus attributes this difference to a different texture of the particles close to the nucleus ("fresh" particles with more compact structure) and further in the coma ("older" particles with fluffier structure). However, as shown in Table 4, a high value of the inversion angle () can also be produced by very compact particles. Indeed, for our samples of olivine we find a value of degrees and it is even higher for the Allende meteorite particles i.e. degrees.
Observational data on comets (Levasseur-Regourd et al. 1999a) show that the degree of polarization (at a fixed phase angle, namely 73o) slowly increases with increasing distance to the nucleus. According to our results, these differences could be due to different size distributions (smaller particles at positions far from the nucleus) or, a more probable option, due to differences in the color of the particles. Particles at positions far from the nucleus could be darker due to thermal processing after perihelion passage and/or because of their interaction with cosmic rays.
© European Southern Observatory (ESO) 2000
Online publication: August 17, 2000